470 ON THE SUMMATION OF SERIES, 
sively, and multiply the resulting equations by 
A, B, C, D, &c., &c., respectively, in order to 
their steps from the solution to the original equations, that 
these developements, themselves, have been obtained by 
either Maclaurin’s or Taylor’s theorems. 
Therefore, to make such an inference as the one above 
alluded to, amounts to no more than saying, that 2, is only 
a particular case of 2”: but who would presume to interpret 
a” without first having a knowledge of w. 
The theorems of Maclaurin and Taylor, are only identities, 
whereas, those of Lagrange and Laplace, are the solutions of 
two important functional equations : this distinction between 
these justly celebrated theorems, will be of greater impor- 
tance as our knowledge of functional equations becomes more 
complete. 
The late very distinguished mathematician, R. Murphy, 
obtained the solutions of the above equations in a simple 
manner, by means of a theorem, which he gave in a memoir, 
printed in the Cambridge Philosophical Society (See Murphy 
on the theory of Algebraical Equations, page 77). 
Professor De Morgan, a great authority in these matters, in 
reference to this theorem of Mr. Murphy, remarks, that it 
“is one of the most general and interesting contributions 
which analysis has received for many years.” (See Diff. and 
Int. Calculus, page 328). 
This may not be an improper place to give a demonstration 
of Taylor’s theorem, without employing the usual supposition 
of an infinite series. 
Let f(2) be the function which we have to develope, when x 
takes an increment h. 
